Now define wn=zn-1 (the sequence of multiplicative inverses of the z's). F is compact, so there exists a convergent subsequence of wn: say v = limk→∞wnk exists, for some increasing sequence of indices nk. By continuity (recall that znk→0),

0*v = limk→∞ znk*wnk =
limk→∞ 1 = 1

so 0*v=1. But this cannot be! (For instance, we deduce that 1=0*v=(0+0)*v=2*(0*v)=2, which is false).

So our assumption that F is infinite cannot hold; a compact field F must be finite.